Integrated research of the general hypergeometric systems and nonlinear integrable systems

本研究的目的是1)研究一般超几何函数和大久保系统，2)研究包括Painleve方程在内的非线性可积系统。GL(N，C)的正则元的中心化子的共轭类由N的划分决定，一般超几何函数是由中心化子的泛覆盖群特征标的Radon变换得到的Grassman流形Gr(n，N)上的函数.我们明确地确定了与一般超几何函数的积分表示有关的代数de Rham上同调群。这个问题已经在n=2的情况下得到了解决，在n&gt；2的情况下，分区(1，…、1)、(N)。对于分区的情况(Q、1、…，1)，证明了上同调群的纯洁性，确定了上上同调群的维数，并给出了它的一个显式基础。这一结果将对构造刻画一般超几何…的Gauss-Manin系统具有重要意义更多的c函数。在分拆为(N)的情况下，我们构造了de Rham上同调的交理论，并用斜Schur多项式来表示交数字。在这一计算中，我们认识到平坦基类比在奇点理论中起着重要的作用。对于P^1上不带附加参数的Schlesinger型微分方程解，利用大久保系统的相应结果证明了其解具有积分表示.这种积分表示是GKZ超几何函数积分表示的特例。对于Painleve方程，我们证明了每个Painleve方程的初始条件空间都存在辛结构，并证明了初始条件空间的几何关系决定了Painleve方程。我们发现了一个有趣的现象，即Painleve II的有理解列的母函数与由Aary函数得到的函数在无穷远处的渐近展开式重合。较少

The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometri … More c functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities.For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function. Less